nLab
coreflective subcategory

Context

Category theory

Notions of subcategory

Modalities, Closure and Reflection

Contents

Definition

A coreflective subcategory is a full subcategory whose inclusion functor has a right adjoint RR (a cofree functor):

CRiD. C \stackrel{\overset{i}{\hookrightarrow}}{\underset{R}{\leftarrow}} D \,.

The dual concept is that of a reflective subcategory. See there for more details.

Properties

Theorem

Vopěnka's principle is equivalent to the statement:

For CC a locally presentable category, every full subcategory DCD \hookrightarrow C which is closed under colimits is a coreflective subcategory.

This is (AdamekRosicky, theorem 6.28).

Examples

  • the inclusion of Kelley space?s into Top, where the right adjoint “kelleyfies”

  • the inclusion of torsion abelian groups into Ab, where the right adjoint takes the torsion subgroup.

  • the inclusion of groups into monoids, where the right adjoint takes a monoid to its group of units.

References

  • Robert El Bashir, Jiri Velebil, Simultaneously Reflective And Coreflective Subcategories of Presheaves (TAC)

Revised on September 8, 2014 21:59:13 by David Corfield (91.125.67.210)